Applicative computing. Its quarks, atoms and molecules.

Library of " JurlnfoR"

Series: Fundamental Basics of Information
Technologies

V. E. Wolfengagen

Applicative computing. Its quarks, atoms and
molecules. Edited by Dr. L.Yu. Ismailova. — Moscow:
"Center JurlnfoR", 20 1 0. — 64 p.

This work covers the advanced topics in main ideas of computing in general.
This material is approved in practice of NRNU MEPhI, MIPT and several
other educational centers of the Russian Federation.

Its 1st part represents an outlook of computations, which is achieved by
adoption of the atomic doctrine for specified reference system of primary
objects. The main attention is given to finding-out of technological features of
computations with objects. Their interaction is considered in applicative
environment that allows finding out internal structure of usual operations
which knowledge allows understanding their properties. The choice of initial
constant entities, considered as primary and referred as combinators is
discussed. These initial entities are used as the basic "building blocks",
entering in applicative environment in interaction with each other. This
interaction results in the constructs, giving representative sets of usual
operators and to the embedded computing systems.

The 2nd part gives some supply of environments for educational and
methodical complex of corresponding discipline (EMCD).

This material is suitable both for advanced learners and beginners in
Computing and Information Technologies as well as in Discrete Mathematics
(DM) and Fundamental Basics of Information Technologies (FBIT). It helps
for developing the intuition sufficient for successful navigation across the
dramatically changing world of innovative information processes which
occurs both in nature and technology.

Material is especially useful for the instructor, postgraduate and graduate
students of IT-specialties and is suitable for the system of training and
advancing the qualification of specialists.

http://www.jurinfor.ru

V. E. Wolfengagen

Applicative computing

ITS QUARKS, ATOMS
AND MOLECULES

a

I

K)pMH*oP

V.E. Wolfengagen

Doctor of Technical Science, professor,
ACM Senior Member

ViacheslavWolfengagen received his Candidate of Technical Science
degree in 1977 and the Doctor of Technical Science degree in 1990
from Moscow Engineering Physics Institute. He is a full professor of
theoretical computer science and discrete mathematics at the
Cybernetics Department of NRNU MEPhI and at the Faculty of
Innovations and High Technologies of MIPT. Since 1994 he has been
with the Institute for Contemporary Education "JurlnfoR-MGU" in
Moscow where he is currently a head of the Department of Advanced
Computer Studies and Information Technologies.

He chaired the 1999-2009 International Workshops in Computer
Science and Information Technologies (CSIT). He is author of the
books Logic: Techniques of Reasoning (2001, Center "JurlnfoR"),
Constructions in Programming Languages: Methods of Description
(2001, Center "JurlnfoR"), Categorical Abstract Machine:
Introduction to Computations (2002, Center "JurlnfoR"), and
Combinatory Logic in Programming: Computations with Objects
through Examples and Exercises (2003, MEPhI - Center "JurlnfoR").

His research interests include data models, database design, software
development databases, object and object-oriented programming and
design, computation theory, programming languages, applicative
computational systems. He was a manager of research and
development projects Logical Applicative Modeling Base of DAta
LAMBDA (version 3, project 93-01-00943 granted by RFBR),
Categorical Object-Oriented Abstract Machine COOAM (project 96-
01-01923 granted by RFBR), Metadata Objects for Proxy-Based
Computational Environment (project 99-01-01229 granted by RFBR).

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V. E. Wolfengagen

Applicative computing

Its quarks, atoms
and molecules

.r JI

Moscow

'Center JurlnfoR" Ltd.

2010

LBC 32.97 Library of "JurlnfoR"

UDC 004 Founded in 1994

B721 Series: Fundamental Basics of Information

Technologies

V. E. Wolfengagen

Applicative computing. Its quarks, atoms and molecules.

Edited by Dr. L. Yu. Ismailova. — Moscow: "Center JurlnfoR", 2010. —
62 p.

This work covers the advanced topics in main ideas of computing in
general. This material is approved in practice of NRNU MEPhI, MIPT and
several other educational centers of the Russian Federation.

Its 1st part represents an outlook of computations, which is achieved
by adoption of the atomic doctrine for specified reference system of primary
objects. The main attention is given to finding-out of technological features
of computations with objects. Their interaction is considered in applicative
environment that allows finding out internal structure of usual operations
which knowledge allows understanding their properties. The choice of initial
constant entities, considered as primary and referred as combinators is
discussed. These initial entities are used as the basic "building blocks",
entering in applicative environment in interaction with each other. This
interaction results in the constructs, giving representative sets of usual
operators and to the embedded computing systems.

The 2nd part gives some supply of environments for educational and
methodical complex of corresponding discipline (EMCD).

This material is suitable both for advanced learners and beginners in
Computing and Information Technologies as well as in Discrete Mathematics
(DM) and Fundamental Basics of Information Technologies (FBIT). It helps
for developing the intuition sufficient for successful navigation across the
dramatically changing world of innovative information processes which occurs
both in nature and technology.

Material is especially useful for the instructor, postgraduate and graduate
students of IT-specialties and is suitable for the system of training and
advancing the qualification of specialists.

NRNU MEPhI • "Center JurlnfoR" Ltd. • MIPT

Aif^H

Part 1
Quarks, atoms, molecules of computing

Abstract. Computing and its development sets up a range of
questions on the most part of which answers either are incomplete,
or unknown. Some of them: what is a 'computation'? What is an
'information'? What is possible to learn, using computing? What
cannot be learned, using computing? - have fundamental value.
In the present work the main attention is paid to finding-out of
technological features of computations with objects. Their interaction
is considered in applicative environment that allows to find out
internal structure of usual operations knowing which allows to
understand their properties. The choice of initial constant entities,
considered as primary and referred as combinators is discussed.
These initial entities are used as the basic "building blocks", entering
in applicative environment in interaction with each other. This
interaction results in the constructs, giving representative sets of
usual operators and of the embedded computing systems.
About the author. Prof. Wolfengagen V. E. (vewQjmsuice.msk.ru),
the head of ACS & IT dept. at "JurlnfoR". He is working in an
area of computing science and information technologies, including
applicative computational systems, A-calculus, combinatory logic,
type systems. RFBR's projects 93-01-00943-a (LAMBDA), 96-01-
01923-a (COO AM), 05-01-00736-a.

Introduction

Computing and its development puts a lot of questions, on
the most part of which answers either are incomplete, or
unknown. Some of them: what is a 'computation'? what is
an 'information'? what is possible to learn, using computing?
what cannot be learned, using computing? - have fundamental
significance.

These questions accompanied computing, since 1940th.
It seemed, there are answers on them, but today the same

http://www.jurinfor.ru J

questions are also indicated by all and everywhere, in all areas
of a science, engineering, business and even policy.

Long time there was a tradition according to which
computing was considered as a science about the phenomena
accompanying computers, and this sight did not raise the
doubts. Computing always was and remains a science about
information processes. Starting approximately with 1995,
experts from different areas of a science, one behind another,
began to declare, that they, in their area, find out natural
information processes. These openings have introduced other
tradition according to which computing began to be considered
as a science about both natural and artificial simultaneously into
use.

By old tradition computing was most naturally described
by ideas from basic technologies - programming, graphics,
networks and supercomputing. The present tradition urgently
demands to express computing in terms of fundamental
principles or even to deduce computing from some fundamental
principles. If one deduced computing from principles not only
its deep structures will be opened, but also their applicability
in other areas of a science will be cleared up as well.
Thus the general aspects of distinct technologies also are
opened, creating opportunities for innovations. At the same
time essentially new ways of stimulation in many respects lost
interest to computing among youth are opened.

In 1940th the computations were considered simply as a
tool for decision of the equations, decodings of codes, the
analysis of data and management of business processes. But
in 1980th computing have developed up to such degree that
have turned to a new scientific method, connecting traditional
understanding of a theory with experiment. And in 1990th
a shift in understanding of a role and a place of computing
followed, as many researchers in different scientific areas have

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come to conclusion, that they have collided with information
processes in natural science deep structures. For example,
these are quantum effects in physics, DNA in biology, thought
processes in cognitology, streams of information in economy.
Computations were included into lives together with new ways
of a decision of the problems, new forms of art, music, cinema,
and also together with new forms of the commerce, new
approaches to training and even new slang on which began to
speak.

The fundamental questions designated right at the beginning
of applying computing became important and in a variety
of those areas in which people in the work essentially lean
on computation and computing methods. Actually, studying
of aspects of computing, bringing the greatest advantage in
traditional sciences, most of all helps an establishment of
fundamental bases and principles of the computing.

Metaphorical speech turns of daily speech have replenished
with phrases like: "I am programmed on such behavior" or
"my brains failed and need to reboot". Active germination
of computing in all the spheres of a daily life has led even
to that from students of the Washington university began
to demand "fluent possession of information technologies",
that assumes their good knowledge and skill to apply in
various situations. There was a need and requirements to
"computational thinking". This assumes usage of principles of
computing both in a science, and in a life. Computations have
got everywhere, and also began to be found out everywhere.

At an establishment of principles and explanations of the
event, proceeding from them, there is one more advantage in
comparison with technologies: it is easier to learn the principles,
than the technologies. The description of a field of activity
in the language of technologies well worked earlier when
the amount of known base technologies was not so big. For

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example, the Association of Computing Machinery in 1989
totaled 9 base technologies, and in 2001 their amount became
already 36, forming 630 direct interrelations that became
problematic for direct studying on former educational patterns.
For today while it is necessary to ascertain, that computing
have not became to be expressed in terms of fundamental
principles, it still remains prescriptional and technological. In
other sciences there was other situation, in them the circle of
the phenomena is established, proceeding from the established
base principles. It testifies to a known maturity of such sciences,
but not computing which only just reaches a status in its growth,
being in which it is possible to speak about its principles.

1 Invariants of computing

The establishment of principles of computing represents an
uneasy process at all. When they in computing were accepted
to business and it began to be developed intensively then
its essence seemed by itself understood, but its forms were
produced. The boom which has arisen in 1970th of the models
of computations does not stop on present time. In the beginning
of this boom nothing constrained the developers, and opening
opportunities promised boundless prospects.

Nevertheless from time to time arose and there is a question
as computing is arranged, but from it more often simply waved
away. Similar interest looks purely academic, and according to
occurring opinion, information and computing technologies are
a destiny of greater companies and powerful collectives. But
the general picture of computing only will win, if it will be
possible in it to find out those invariants, which are kept from
change of forms.

Invariants play a role of global constants, and from the
computing point of view - primitive 'building blocks', using

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with which it is possible to design this or that 'world' of
computing. As on each of available forms of computing
its developers precisely and rigidly declare the rights, both
developers and users zealously defend the world designed by
them. They concern to this world, as by environment of the
information dwelling, stopping attempts of intrusion into it from
the outside and if it was possible to establish unity of forms it
would promote improvement of mutual understanding in the
diversity of information community.

It was known that similar invariants exist. The question
is put differently, as how to use them to take advantage.
Really, it is necessary to recollect about combinators, rather
recently opened in the world of metamathematics, as it supports
confidence of an available unity of forms. From the intuitive
point of view an environment of computations contains all or
nearly so everything, that concerns to construction of a result:
there are variables and their actual values in it, and they are
carried not only positionally, but also contextually

Everything that is done in computing, can be dropped
down to some fundamental principle which many admit, and
which is not rejected by overwhelming majority: consider the
identifier and relative to environment associate for it some
construct which will be considered as its value. Computation
considers this process of constructing, and computing develops
technologies of implementation of the construction. So,
the correspondence between the identifier and its value is
parameterized by environment. The number of all identifier-
value possible combinations is so great and impressive, that
it is not feasible to construct the hypothetical table, using
which all the possible instances concerning the identifiers and
their values are listed. Certainly, for realization of similar
strategy of computation one should get a huge database which
would be in a status of permanent upgrade and updating. At a

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modern level of understanding it is considered technologically
unacceptable. From the positions of a theory of databases
and relational model this relation is considered as explicit
transfer of all possible combinations of individuals in which
they can appear, remaining within the limits of this relation.
From a mathematical point of view the point under discussion
is a mapping for which are in advance known the range
of definition - its domain, - and the range of value - its
range. They are not only great in volume, but also are subject
to changes, and development of a theory of mappings with
variable domains-ranges is in embryo status. D.Scott in (D. S.
Scott, [17]) has suggested to consider constructions of variable
domains, but in the field of computer sciences it was not
widely supported, as burst a little bit later the boom of
computing prompted other promising prospects at once in many
directions, and for reception of fruits it was not necessary to
bring a burdensome payment of development the complex and
interdisciplinary theory, which efficiency should be defended in
addition.

In combinatory logic argued differently, believing, that it is
necessary to borrow in designing actual mappings, not caring
about existence of their ranges of definition and ranges of
value and due to efforts of M. Schonfinkel (M. I. Schonfinkel,
[15]) and H.Curry (H. B. Curry, [6]) such a theory has been
developed. More radical intention consisted of finding the
minimal set of mathematical entities, using with which it would
be possible to design all building of modern mathematical
knowledge as other forms of knowledge were not considered
but believed having no right on that with them seriously were
considered. It is important, that opened by them combinators
underlay any mathematical and metamathematical reasoning.
Later, in a process of developing the information technologies,
their fundamental value for computing has been known.

o http://www.jurinfor.ru

In traditional computing a central concept, without which
as usually it is considered, it is impossible to operate, is
the representation of a variable. The variable plays a role of
'numbers in general', that helps to build the general statements
and to analyze their properties. At once it has been realized,
that variables should be considered more widely, including
the 'entities in general', some indeterminants, not reducing
their sphere of action only to pure arithmetics. Combinators
are perceived as 'constants in general', assuming, that the
structure of knowledge, by its organizing, is granulated by
constants. At the same time it is not unexpected, that they either
in mathematics, or in computing have no reliable definitions
neither of a variable, nor a constant. Such situation undermines
trust to the fundamental trues saved up in these areas which
formal expression with necessity is grounded both on constants
and on variables.

A special attitude to variables as to constructs which
in any way are impossible to be avoided, has generated
the modern reality of information technologies expressed in
programming languages. Predilection for variables generates
difficulties of basic nature when the question of application
of computing arises. Fitting the knowledge within a form is
carried out in books, for which the way of expression and
organizing is considered so well-known, that, actually, is not
exposed to studying. An exception is the project Automath
by de Bruijn (N. G. de Bruijn, [8]). Having started in 1967, it
pursued the aim to develop the environment for expressing the
mathematical theories in a form suitable for computer check of
their correctness. A hypothesis, that if a statement is expressed
correctly then it is correct in fact, was laid down in its basis.
Any other norms of correctness were not introduced. There is
no need to forget, that 'correct' means simply 'based on rules'
and it remains to understand just a question, what a rule is,

http://www.jurinfor.ru y

but this is the most debatable question not only in modern
metamathematics, but also in computing.

It is considered, that a language by its structure forms
logic, that immediately leads to necessity to clarify its
fundamental difficulties which though are known, but are not
considered quite perceived. Presumably neither logics, nor
the mathematical grounds were not used in Automath. In
this project all the mathematical material formed the books,
written in a language, and the language was based on lambda-
calculus with types, in terms - in the rules, - of which
representations about 'definition', 'theorem', 'proof, 'axiom'
were expressed. The book represents a construction consisting
of nested blocks, and opening of the block corresponds to
introduction of a typed variable. Variables present mathematical
objects or mathematical proofs, and the system of their mutual
correspondences is developing quite similarly. In other words,
an idea of proof-as-object was incorporated in this project from
the very beginning.

The only interpretation of this block structure is on a
share of logic metameans. Is it a lot of or not - sets
up an open question. Is it enough to introduce in the
expressed in such a way text the fundamental difficulties
burdening metamathematics? However, by a form of its
expression the project represents an innovative way to consider
relation of logic with mathematics, at least, from positions of
possibilities of modern computing. Processing the books which
are expressing the mathematical knowledge in a language of
Automath, becomes attractively easy and natural for dominating
general mathematical practice. It gives some didactic and
educational opportunities: the teaching of mathematics so that
students could study it, receives a sound ground in the available
and developing information technologies. Moreover, teaching
from a category of art passes into a category of technology

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when it is enough "to explain" the machine - in a language
of the project, - the constructive organization of the text
by its form, but not by its meaning. Thus any preliminary
logic or mathematical "implied sense" is not brought in, and
the validation of correctness of the text is assigned to the
environment of computing. On a plan, this approach does
not cover the automation neither of a mathematical invention
process, nor of search of the proof process for the theorems
having been formulated: Automath plays a role of the attentive
reader of the material which is correctly represented by its form.
As it has appeared, it was required to develop the virtual
reader in computing which will correctly carry out a process
of reading the virtual book which is correctly applied by
the appropriate form. It is caused by extreme complication
of mathematicalized knowledge when its mastering or
estimation of correctness exceed usual human abilities. But
not possibilities, as, taking to the aid computing and its
environment, knowledge turns in from actual to possible one. In
the newest information technologies it has renewed heightened
interest to semantic networks (T. Berners-Lee et al, [3]).

2 New paradigms of computing

The sense of a term 'computing', maintained nowadays though
with changes, but corresponds to that understanding which
has been accepted 60 years ago at establishing of the ACM
(Association for Computing Machinery). The shifts which have
outlined at the newest time in its understanding appear rather
essential, having three general characteristics: computing is
not necessarily carried out only on the basis of technology
of silicon integrated schemes; the basic computing elements
are implemented physically, becoming not simply a theory;
transition to the new thresholds of miniaturization often making

http://www.jurinfor.ru 1 1

from 1 up to 100 nanometers is carried out. Under these
conditions the customary representations about computing start
to be reconsidered, but this does not mean at all refusal of
available representations or technologies.

Some of new forms of realization of computing have
the expressed addressing and are intended for the decision
of quite certain problems, for example, having the raised
computing complexity or concerning particular applications.
Though practical and daily application of a majority of them
is only ahead, expecting occurrence of suitable devices, their
modeling originality carries away its natural fundamentality,
innovation and potential. The opportunity to create a basis
for new forms of processing of the information opens in the
latter case and, being based on them to develop families of
applications. Discussion of their technological opportunities
occurs usually outside of sphere of the periodic literature on
computer sciences, and research is moved to area of such
natural sciences, as physics or chemistry. It is caused by a status
of works which for the present time are at a level of study of
basic ideas - or in the most initial technological phases, or at
a level of experiments. In a process of technological ripening
similar forms of computing get in a sight of computer sciences,
starting to be used in practice. Realization of new forms of
computing encounters difficulties of development of hardware,
demanding development of new architectures. On the other
hand, difficulties are caused also by attempts to equip new
forms of computing by the suitable software as it is required to
develop new schemes of the organization of computations and
new algorithms. The situation is similar to that which was at
transition from consecutive algorithms to algorithms for parallel
computing systems. Some known decisions can be transferred
on a new area, and others - cannot.

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The known new forms can be subdivided into two greater
groups. In the first of them are put the decisions based on
nanotechnologies . They are the organization of schemes of
computations on nanofibres, coal nanotubes, organic molecules,
bio-DNA and quantum effects. In the second are put the special
forms of computing including optical, micro/nano liquid and
chaotic computations.

3 Revision of computing foundations

The consumers in a customary computing have promoted in
understanding of sets and have learned to maintain the models
of computations based on the notion of a variable for which
it is known, what domain it will range - typed models of
computations, or models of computations with types. In other
words, an idea of type has received a wide circulation and a
universal recognition, and all available programming systems
have to more or less extent worked out management systems of
types of variables/objects.

The models based on classes remain less worked out as
they conduct to construction of domains which elements are
other domains in turn etc., and for such structures the volume
of computations needed to evaluate true or false of statements
sharply increases.

The models of computations, in which indication of
type of variables was not supposed - the untyped models
of computations, remain completely neither worked out nor
comprehended in practice. The leading position among them
is borrowed with A-calculus and combinatory logic. Though
A-calculus also is recognized in practice of programming
technologies, but not so fast, as it deserves that, the combinatory
logic is applied obviously insufficiently. Combinators - the core
primary elements of combinatory logic, - were introduced in

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hope to get rid of arithmetic style of working with numerical
data, characteristic for overwhelming majority both of former
and existing programming systems, and instead of it to pass
to other style of reasonings in terms of objects and their
applications to each other. In use of the first style the
calculation is carried out - from a words 'to compute numerical
value', - and in use of the second one - computing in the
true and self sense of this term. Besides that combinators deal
with/ree variables which are understood as indeterminants and
have no deal with bound variables at all. As a matter of fact,
computing with combinators is carried out in terms of constants.
It is even better to say: 'constant objects', and they are constants
not in absolute sense, but in a relative sense when objects
reveal the property of a constancy relatively the environments
in which computing is carried out. It remains to formulate
suitable definition of a constant - to give the characteristics to
constancy property, - and also to be determined with a model of
environment. This just appears uneasy business as influences all
the elements of computing architecture without any exception.
Now we approach to an important point - necessity to
formulate the representation of a constant, which would appear
fruitful for computing in general. Certainly, it would be
desirable to leave this representation both intuitively transparent
and coordinated with an available natural- science representation
of a constant.

4 Notion of a constant

Do we know a lot of or a little concerning what a "constant" is?
To the essence, everything connected with representation about
a constant has appeared subcontracted to the mathematics. And
both representation about a constant, and representation about
a variable are considered as self evident in it. At the best it

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admits, that the constant is - unlike a variable, that does not
vary, and the variable, in particular, can be a constant. These
representations remained firm or seem those till a time. So
would be now, if not computing.

It has appeared, that the best that was offered is a general
agreement on a constant which other side was a variable.

At a discussion, discourse is usually conducted about values
behind which numbers are there and then seen. Numbers and
their processing - calculations, - worked for our advantage
for a long time perfectly and trouble-free. Even in computers
while their productivity has not grown so, that the technology
has approached to the physical threshold of miniaturization
allowing precisely fix and distinguish zeroes (0) and ones (1).
And now there comes some critical point in understanding,
what the computing is.

Whether is there a theory of constants? The answer to
this question is known, even more: it appears, that there
is no language on which it would be possible to speak
about constants mathematically precisely. Attempts to speak
about them in absolute sense look restricted and insufficiently
grounded. In a relative sense it is possible to achieve greater:
why not to name a constant the 'object' which does not depend
on certain 'context'? Or, even better, it does not depend on
'environment' which is considered as some context for a while.
We shall look, whether is there an adequate language for exact
expression of this idea.

For a test we shall take A-expression

(Ax. object) (environment) = object,

which, obviously, expresses the thesis of constancy: (Ax. object)
can be considered as a unary constant function. But transition
to A-language has demanded introduction in a consideration the

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representation about a 'variable' which also appears bound by
the abstraction! In this moment all power of usual mathematics
is received, but simultaneously all the known costs of operating
by variables is inherited.

Let's take expression of combinatory logic

K (object) (environment) = object,

to which should operate under the scheme of axioms K :
Kab = a. To say in a language of combinatory logic, that
the object is a constant one, that represents a constant relatively
to environment, means, to apply the combinator K to it. Then
- relatively to the given environment, and this is, possibly, the
other object, - the 'object' is simply quoted, that now quite us
arranges.

How precisely is to tell in mathematics, that the object
varies? For this purpose we shall write down

^(object) (environment) = object',

where V is some combinator which acts on the object
interesting to us - old object, - relatively the given context
and results in a new object - object'.

And again, it is necessary to bring in a traditional
mathematics to record this idea in the A-language. We write

F (Ax. object) (environment) (object- 1) =

V (object) (environment) x:=0 bject-i,

that is, the old environment has appeared to be replaced by
a new one (environment) ;E:=0 bject-i, differing from the old one
unless that, instead of a variable x, substitution of the object
'object- 1' is executed. There was generated an expression with

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a variable which has to be served by a newly introduced rule of
substitution with all the known consequences.

Thus, applicative on its plan A-language appears burdened
by all known technical difficulties of processing both free and
bound variables. In a language of combinatory logic such a
burden is not still present.

5 Functions

The usual idea, concerning functions, consists of that they
are a special case of a law of correspondence putting in
predetermined relation to the elements of one domain the
elements from other domain. So first of all it is necessary to be
determined with a domain A which is considered as represented
by a constant object, that is its structure obviously does not
include free variables. For domain A and combinator B we
shall demand

A = Ao A = BAA = 1 A .

For mapping / : A — > B where / is an object which does not
contain free variables, we shall demand

f = BofoA = Bo (BfA) = BB(BfA).

Thus, mappings / are determined as the triples (A, f, B).

6 Interaction of an object and environment

6.1 Environment

A thesis, that interaction of objects needs the intermediary -
environment, is perceived as obvious one. At least, currently
it does not attract doubts. More rigorously, to initialize an

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interaction of objects, the structure is needed where they are
localized.

Opposite case - when some "wandering" objects "meet"
other wandering objects, - is interesting, but this discussion
will be postponed for a while. The area of programming gives
a case when objects, by some way or otherwise, are already
packed by in the environment. Thus a central concept under
development is namely the environment which is understood
as an environment for computations. Environment is equipped
with the programming system, but not wise versa.

Other circumstance is that an object interacts not with all
environment at once, but with its partition - that which will
appear "in an area of action" of the object.

Prestructure. An applicative prestructure is used for packing
objects. Two aspects of an object - its redex (reducible
expression) and the contract, - reveal in it. In other words, the
prestructure gives a representation of computation both in terms
of a reduction - transition from redex to the contract, - and in
terms of expansion - transition from the contract to redex.

The principle of interaction gives some non-symmetry: there
is an object-initiator of action and there is an object- recipient
of action. Influence of one object on another is stepwise: it
is carried out, if and only if objects are located immediately
beside. The arrangement happens of two kinds: beside and
not beside (distant), and in the second case the objects do
not interact. In case of an arrangement beside, the objects
immediately enter in interaction. The new object, as a result
of interaction, arises and begins its existence - result of acting,
or applying of the first object to the second. Now, if there will
be an object located beside thus newly born object, the new act
of interaction begins where are two distinct cases.

1 o http://www.jurinfor.ru

In the first of them newly generated object captures the
existing one, which has appeared beside and acts on it.

In the second case newly generated object is captured by
the existing one which affects this object.

In any of these cases the new object arises and begins
its existence and this object is considered as a result of such
non- symmetrical interaction of two objects-parents. It settles in
prestructure on the equal rights with other objects. In particular,
this means the following: as soon as the new object-result
is generated, it is possible to speak about the new act of
interaction.

Thus, the inhabitants of prestructure participate in
interaction which evolves by a principle of a dominoe. The
following circumstance is important: either there are initial
atomic objects, or there are derived non-atomic objects, each
having exactly two ancestors-parents. A question still open
where are the initial objects from, but this discussion will be
postponed for a while.

6.2 Interaction

The object can be reveled in interaction with other objects if it
participates in application. In this case it can show arity, equal
to (constant object) or distinct of zero. For simplicity we shall
consider a case when the object shows arity, equal to 1 (unary
function).

As interaction is carried out through the intermediary -
environment, - then some metaoperators will be required. For a
while, we shall be limited by two metaoperators: A - currying
and || • || • - evaluation map. For any object M we shall check
up, whether it can show arity 1 in the environment i. To obtain
this we write down

||M||id ,

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which represents a value of object M in the environment i. If
value of object M shows arity 1 then there is a construction of
value of object in the environment

A\\M'\\id ,

where M' is the same as object M everywhere, except for a
variable to which we should assign the value d : instead of this
variable, the number of de Bruijn is written as a prototype
of a pointer to d in environment i'. Environment i' is the
same as environment i everywhere, except for an image of this
substitutional variable, which is now assigned d :

\\M'\\[i,d ].

Actually, it was necessary to create a compound metaoperator

A\\ ■ || ■ : object x environment — > value,

which is an object generating, setting up the function of arity 2.
Really,

A\\M'\\id = \\M'\\ Mo]-
=*'
For example, if M is an identity transformation I with the
characteristic Id = d then it is sufficient to assume, that M'
is a substitutional variable which is assigned the value d in
environment i:

||/||*rfo = -^|| 1| i do
= Pll [hM

= Snd[i, d ] = d ,

as was expected. Here vl||0||z is an image of object I, obtained
as a result of its interaction with environment. This should

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be simply a pointer Snd to d , located in the modified
environment.

Other example. If M is a cancellator K with the
characteristic Kdid = di then it is sufficient to assume, that
M' is a substitutional variable which is assigned the value d 1 in
environment i:

\\K\\idido = A(A\\l\\)id 1 d
= A\\l\\ [i, di] do

i'

= ||I|| [[i, di],d ]

i"

= (Snd o Fst)i"
= Sndi' = di,

as corresponds to the characteristics.

And one more example. If M is the allocator S with the
characteristic Sd 2 did = d 2 do(dido), then

\\S\lid2dido = yl(yl(i4||20(10)||))id 2 dido
= yl(yl||20(10)||)[i,d2]dido

= ^||20(10)|| [[i,d 2 f,di] d
= ||20(IO)||[[[z,d 2 ],d 1 ],d ]

V v '

»'"

= ||2||i /// (||0||i /// )(||l||« /// (||0||« ,,/ ))

= Snd o Fst o Fst i'"(Snd i'")(Snd o

o Fst z'"(Snd i'"))
= S , ndoFst« // d (S'nd« // do)
= 5nd i'do(dido) = d2do(dido) ,

as was expected.

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7 The principles of computing

Setting up the principles of computing, it is necessary to accept
the assumptions, and as it appears, some of the fundamental
premises, probably, by virtue of their seeming simplicity and
deceptive self-evidence, escape attention of the researchers.

One of them and, possibly, core premise, is in acceptance
of applicative structure as an environment of embodiment of
computing. It means, that some entities/objects are acting,
or applying to others. For example, the object-function is
applied to object-argument, and the result of this application
is considered as a value of the given function of the given
argument. All of this looks quite obvious, but is almost
never formulated explicitly, resulting in various displacement
of accents.

Another - and again all known, - assumption can be
dropped down to a simple formulation of operating with
identifiers: given that is considered as an identifier, and
for this relatively the environment is constructing that will
be considered as a value. This process of constructing is
considered as a computation (in a sense of evaluation),
and computing develops technologies for realization these
constructions. Thus, the relation between the identifier and its
value is parameterized by an environment.

In the environment not all the objects are isolated, an
interaction of the objects is the most interesting, but it
proves through application. This is a structural metaoperation
which operates an interaction of objects and it would be
undesirable, that during performance of computation of value
of the identifier its own properties have been changed.
Hence applying will be considered as invariant relatively
evaluation, and this property needs obvious characterization by

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acceptance of the general principle of evaluating the application
(V. E.Wolfengagen, [22]).

Principle of evaluating the application. The base premise is
as follows:

evaluation of application is the application of evaluations.

The formulation above needs augmentation because of
evaluation can be executed both under fixed and unfixed
environment. In the first case when the environment i is
assumed as fixed suppose:

\\MN\\i= (||M||i)(||iV||i)

for arbitrary objects M, N. Then by purely formal reasons,
following from the general properties of computations,

(||M||z)(||iV||z) = (Xr.r\\M\\ \\N\\)Si
= CIS(Xr.r\\M\\ \\N\\)
= 5[||M||,||iV||]i

for S = Xxyz.xz(yz), C = Xxyz.xzy, I = Xx.x, S = CIS
and ordered pair [x, y] = Xr.rxy. Hence, the following rule

(rule 1) \\MN\\ = <S[||M||, ||iV||]

can be formulated, which is derivable in A?]^ -calculus.

Principle of evaluating the ordered pair. The main premise
is formulated as the equality

evaluation of ordered pair is the ordered pair of evaluations.

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Formally this equation is rewritten as

||[M,iV]||z = [\\M\\i, \\N\\i]

for any terms M, N and assignment i. By the postulates (77),
(£) and (r) of A-calculus, the following rule is derivable from
the main principle:

(rule 2) ||[M,iV]|| =< ||M||,||iV|| >,

where < /, g >= Xt.[ft,gt] for arbitrary /, g. Both the
principles above and derived rules make it feasible to obtain and
ground the standard semantic features of computational models.
As the most important for the means of conceptualization we
indicate two corollaries which correspond the evaluation of A-
expressions and applications as is written below.

Evaluation of A-expression. Assume the application (X.M)d,
where M, J are any terms, and (A.M) denote the abstraction of
(some) variable. Then the following consequence of equalities
is valid:

||(A.M)d||i= (||(A.M)||z)(||d||z) (by principle 1)

= (\\\.M\\i)d (assuming (||d||i) = d)

= A 1 1 M 1 1 id (by definition of A)

= \\M\\ [i, d] (by Ah = Xxy.h[x, y\).

Hence, the rule:

(rule 3) \\(X.M)d\\i=\\M\\[i,d].

is valid as well. In the following we will accept that this rule
determines a generating function: evaluation of M "generates"
the replacements by d matching M.

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Second way of evaluating the application. In evaluation of
application the definition of applicator e:

\\MN\\i = (||M ||i)(||iV||i) (by principle 1)

= e[||M ||i, \\N\\i] (by definition of e)

= (eo < ||M||, ||AT|| >)i (by definition of < ■, • >)

is used. This sequence of equalities leads to the rule

(rule 4) \\MN\\= eo <\\M\\,\\N\\> .

At last, we indicate the case of evaluation of individual
constants:

||c||i = c,
which (in A^-calculus) results in the rule

(rule 5) ||c|| = Xi.c.

This means that individual constants are independent on the
particular assignment, i.e. they are statical.

List of rules. For reference, a complete list of the derived rules
is given below:

(rule 1)

\\MN\\ =S[||M||,||iV||]

(rule 2)

||[M,iV]|| = < ||M||,||iV|| >

(rule 3)

||(A.Af)d||i= ||M|| [i,d]

(rule 4)

\\MN\\ =eo < ||M||,||7V|| >

(rule 5)

c = Xi.c.

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8 Interaction of objects

8.1 Unfixed number of argument places

First of all it would be desirable do not link the interaction
of objects to traditional mathematical representations. At least,
not to do this at once and without any visible necessity.
The mathematical intuition prompts, that if something acts on
another it is necessary to consider the first as a function, and
the second - as an argument. Under these conditions the result
of interaction is considered as value which, in turn, is an object.
But if there is a function, it is characterized by its arity - by
number of its arguments or, more precisely, argument places. In
the usual mathematics the number of arguments of a function
is known in advance, but in case of acting of one object
on others it is not known in advance, on how many objects
it can act. If the object is considered as a function, then
its number of argument places appears in advance unfixed,
and the object reveals its arity in interactions. Whether such
mathematical functions are known? As it appears they are only
the combinators.

8.2 Object and its sphere of action

Let there is some set of objects arranged in a structure. We
shall start from the point that not any object will cooperate
with everyone. If we speak, that the object a acts on object b,
we have in mind that a is applied to object b.

From Fig. 1 it is clear that in the sphere of action of object
X there are both the objects a, and b, but a does not act on b.
In a mathematical notation it is written down below:

X.. ab.. = (..(((X..)a)b)..).

From Fig. 2 it is clear that in a sphere of action of object X

26 http://www.jurinfor.ru

\

X)..ab..
/

Fig. 1. Object X affects the objects a and b, where a does not
affect b.

Fig. 2. Object X acts on the object (ab), where a acts on b.

there is an object a, which acts on b, so that X acts on a result
of action of object a on object b. Notationally this is written
down as follows:

X..(ab).. = U(X..)(ab))..).

Parentheses in which objects a and b are concluded, are
essential, and they cannot be omitted.

9 System of primary objects

Possibly, it is necessary to make some assumptions. They will
concern presence of some set of primary objects - in fact, it is
necessary to have in stock actual, actually existing objects. At
the same time we shall make an attempt to conduct discussion
of relations between objects, whenever possible, without those
assumptions, in particular, concerning the existence of objects
which can be avoided.

First, an ability to generate any object b is required. Let
it will be always accompanied by occurrence and application

http://www.jurinfor.ru 27

of a constant object K. Other version of this reason can look
differently: it would be desirable to formulate the statement,
that some object a does not depend on environments b. It means
also, that purely by syntax the object K incurs function of
quoting a in an environment or in a context b. Application of
K also means encapsulation of object a within environment b.
Each of these explanatory systems can be used depending on a
context in which studying of objects is conducted.

On the other hand, there is also a symmetric opportunity of
elimination, or the termination of existence - cancelation out, -
any object b. The termination of existence of object b is caused
by the termination of existence of a constant object K, directly
acting on some object a, and the object a remains and continues
its existence.

From Fig. 3 it is possible to understand behaviour of object-

Fig. 3. Characteristic of object K: a = Kab.

cancellator K. On expansion of any object a the object K is
generated, starting its existence, and object a appears directly in
a sphere of its action. In additon the object b is generated which
gets in a sphere of action of a result of interaction of object
K with object a. Upon reduction the cancellator K eliminates
object b which ends its existence, but this instance of K also
does not exist any more.

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At the same time it can be demanded to eliminate clones of
object, leaving only its single instance. Certainly, the symmetric
operation of distribution of actions on various instantiations of
any object c, carrying out their cloning is expected as well.
We shall try to carry this out as follows: on elimination of
a clone of object c the constant object S, of a special kind,
will be generated, and on cloning c - to the contrary, one of
instantiations of object S will be enforced to end its existence.

All of this means, that various instances of object will
be indiscernible in applicative environment. This idea of
indiscernibility of a clone of object is presented at the scheme
of computation which determines a behaviour of the allocator
S: ac(bc) = Sabc. Apparently from Fig. 4, that object S -

Fig. 4. Characteristic of object S: ac(bc) = Sabc.

in applicative environment, - is applied to objects a, b and
c, capturing them. This means also that the arity of initial
primary object-comb inator S equals 3. Its action directly does
not extend on the other objects in the environment. But all the
construction of Sabc is reduced to ac(bc), the object c is cloned,
and computation is distributed: one instance of c directly occurs
in the sphere of applying of a, and its second instance - in the
sphere of b. Computation is distributed, but the result obtained
from interaction b with c, occurs in a sphere of action of the
result obtained from interaction a with c. The most essential,

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that at generation of a new object S: ac(bc) = Sabc the clone
of object c stops its existence, and object S starts its existence,
becoming an inhabitant of applicative environment. This is an
essence of S-expansion. On the other hand, at acting of object
S on kept by it in the environment objects a, b and c, occurs the
cloning of c, this clone begins its existence in the environment,
but at the same time S stops its existence. This characterizes
S-reduction.

10 System of derived objects

Now the way of "detecting" the objects with predefined
characteristics in applicative environment is, in general, clear.

Process of detection of a new object appears rather
constructive: it is necessary to show a construction built from
already found out, old objects. Thus, new objects appear
derivatives while gradually explicate a representation about the
initial objects, all or part of which appears the primary ones.

Let's consider, by example, the synthesis of a new object
with predefined characteristic. Let in Fig. 5 the characteristics
of such an object is as follows: this is the combinator-
permutator C, carrying out rearrangement by places of objects
b and c.

Fig. 5. Synthesis of object C with predefined combinatory
characteristic: Cabc = acb.

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A synthesis of C is carrying out as follows. Starting with
object acb, which by its structure is a result of applying to b the
result of applying a to c.

First, we shall execute cloning of c. For this purpose we
shall make K-expansion on the basis of object b, generating the
second instance of object c, which is possible to see in Fig. 6.
Second, execute 5-expansion on the basis of object ac(Kbc) in

C^\\ exp
ZLS J red

Fig. 6. K-expansion and generation the second instance of
object c: acb = ac(Kbc).

accordance with Fig. 7. During its execution the second instance

Fig. 7. 5-expansion and elimination of the second instance of
object c with permutation of object Kb: ac(Kbc) = Sa(Kb)c.

of object c is eliminated, but the allocator S is generated.
Third, execute S-expansion according Fig. 8.

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31

x i exp
©a (K)bc^z=±
red

Fig. 8. 5-expansion and elimination of a composition of objects

Sa and K: Sa(Kb)c = B(Sa)Kbc.

Fourth, execute 5-expansion, which eliminates the compo-
sition of objects B and S. The same time execute the cloning
of object a, using /-^-expansion based on object K. The new
instances of objects K and a are generated, starting up the
existence in an environment. This transformation is done in
accordance with Fig. 9.

exp
S>Kb[c] ^=±
red

Fig. 9. i?-expansion and elimination of composition of objects
B and S, carried out together with K-expansion based on
K with generation the clone of object a: B(Sa)Kbc =
BBSa(KKa)bc.

And, at last, fifth, execute S'-expansion, which eliminates
the distribution of computations BBSa and KKa together with
clone of object a. The second instance of object a cancels out

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the existence, but the object S - starts up the existence in
environment. This is represented in Fig. 10. Now the desired

exp

a[b][c]^=±(s) ®Bs((K)Ka[b][c;

Fig. 10. 5-expansion and elimination of a clone of the object a:

BBSa(KKa)bc = S{BBS)(KK)abc.

order of objects a, b and c is achieved in environment, i.e.
objects cub have changed their places.

In this process the combinator B with the characteristic
Babe = a(bc) is used. It seems clear that it can be synthesized
as well, using the only combinators K and S. In Fig. 11a

exp,
red

Fig. 11. Characteristics of the combinator B.

characteristic of combinator B is shown.

We will show that combinator B can be obtained by
combining K and S. First of all let's fix the object a(bc).
The idea is to get the object c free of immediate action from
the object b. Mathematically this means the need to omit

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33

parentheses. To get this, the distribution of computations will
be synthesized, generating an additional instance of object c,
that is reached by occurrence of an instance of combinator K.
Write down symbolically:

a(bc) = Kac(bc).

Further, we eliminate one of instances of object c, that needs
the occurrence of an instance of combinator S, but, in passing,
remained instance of c is get out of dependence on b:

Kac(bc) = S(Ka)bc.

Similarly we shall get out now object a of dependence on object
K. It is necessary to do this in two steps. First we shall generate
the second instance of a, having distributed computation and
having generated an instance of combinator K:

S(Ka)bc = KSa(Ka)bc.

In Fig. 12 all the derivation chain is shown as

a(bc) < Kac(bc) < S(Ka)bc < KSa(Ka)bc.

Now one of two instances of object a will be eliminated, for
which it is necessary to generate the object S:

KSa(Ka)bc = S(KS)Kabc.

The target object is synthesized, it remains to assume only that

S(KS)Kabc = Babe.

In Fig. 13 a continuation of derivation is shown, starting with

KSa(Ka)bc:

KSa(Ka)bc < S(KS)K abc = Babe.
= B

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Fig. 12. Derivation of combinator B: a(bc) < Kac(bc) <
S(Ka)bc < KSa(Ka)bc.

c)SaCK)abc< > i (?) (2)SKa b c = ( ((Y§)abc

Fig. 13. Derivation of combinator 5: KSa(Ka)bc <
S(KS)Kabc = Babe.

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35

Thus, the full chain of a derivation looks like

a(bc) < Kac(bc) < S(Ka)bc < KSa(Ka)bc < S(KS)Kabc

B

Babe.

By the same way it is possible to synthesize identity combinator
J with the characteristic la = a, that is shown in Fig. 14. This

exp _

Fig. 14. Characteristic of combinator I: la = a.

is a special combinator, under its action any object a does not
vary, and speaking more exactly, remains self-identical, passing
in itself.

A derivation for combinator I is shown in Fig. 15. We start

GXD / /*~ ~N ""~~ ~~"X GXt)

(a) < ? ((K)a(K)a< ? f i f (s)KK a = f (I) a
w red \ ^>^y W / red '' v ^^

Fig. 15. Derivation of combinator I: a = Ka(Ka) = SKKa
la.

fixing the object a. We generate object (Ka), but this arises an
instance of object K as well, which starts its own existence.

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Now it has appeared, that two instances of object a are available
which structural arrangement allows to eliminate one of them.
But thus the object S, which begins the existence, is generated.
Now it has appeared, that the object SKK by its characteristic
does not differ from the object I, demanded by the synthesis.
So the purpose is reached, and mathematically it can be written
down by means of I = SKK.

Primary and obtained during the formation of objects, and
some of the derived combinators are represented in Table 1.

11 Derivation of combinators

Given such an atomic-and-molecular constructor, it is possible
to synthesize the objects with predefined computational
properties - the combinatory characteristics.

In the Fig. 16 in a scheme for combinator S the object b

Fig. 16. A particular construction of combinator S for b = I.

is replaced by the combinator I. This initiates a reduction in
the direction from Sale to ac(Ic). The last object contains the
redex Ic, which is replaced by the contract c, so that all the
reduction works as follows:

Sale \> ac(Ic) > ace.

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Table 1. Primary and derived combinators.

Combinatory characteristics

Interaction of objects

Kab = a

Sabc = ac(bc)

la = a

Wab = abb

Cabc = acb

Babe = a{bc)

9 abed = a(bc)(bd)

$abcd = a(bd)(cd)

exp \

exp

< J

red

©c ®c

/^\ exp

exp

(w>h < ; (a)bb

' red v

exp

(VVh -z » (C)abc

' red

38

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But inside the object Sale it is necessary to make
transformations: this concerns a position of combinator I.
Transformation will be executed by expansion:

Sale < CSI ac = Wac,
= W

but both the chains of synthesizing the object can be merged:

Wac = CSIac > Sale > ac(Ic) > ace.

A process of synthesis the object W is represented in
Fig. 17. Now the newly synthesized combinator W, which, as

Fig. 17. Synthesis of construction of combinator W with a
characteristic Wac = ace.

appeared, behaves exactly like CSI, can be added to available
combinators. In Fig. 18 a characteristic of combinator W is
given.

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39

exp /7^C\ X
' red Vv^Sv

Fig. 18. Characteristic of combinator W.

12 Reduction and expansion of objects

Once again turn back to features of obtaining the combinatory
representation of object-permutator C. There are in Fig. 19 the

exp,
red

K)b c

K

red

KKa

y red

Fig. 19. Synthesis of supplementary objects: b = Kbc, K =
KKa, B(Sa) = BBSa.

supplementary objects, which are obtained during a synthesis
of target object C with predefined combinatory characteristic

Cabc = acb.

Everything what is required is to change places of the second
and third objects in acb. However, it is necessary to remember

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that rearrangement should be made in applicative environment.
First of arising ideas consists in trying to find transformation
of acb, having executed which it is possible then to apply
transformation under the scheme S.

Such an attempt is reflected in Fig. 6 and led to a need of
generation the additional instance of an object c - its clone,
but not a copy. In other words, distinct instances of object in
applicative environment will be indiscernible. Such an idea of
indiscernibility of clone of object is present in the scheme of
computation, defining behaviour of the distributor S: ac(bc) =
Sabc. Apparently from Fig. 4, the object S - in applicative
environment, - is applied to objects a, b h c, keeping them.

13 Synthesis of an object with the given
combinatory characteristic

Let's return to consideration Fig. 5.

It is required to clone object c, and then to eliminate it,
having taken advantage of ^-expansion.

First, we shall execute cloning of c. For this purpose we
shall make K-expansion on the basis of object b, generating the
second copy of object c, what is possible to see in Fig. 6.

Second, we shall execute S'-expansion according to Fig. 7.

Third, we shall execute S-expansion according to Fig. 8.

Fourth, we shall execute S-expansion which eliminates a
composition of objects B and S. At the same time we shall
execute cloning of object a, using K-expansion on the basis
of object K. New instances of objects K and a are generated,
beginning the existence in the environment. This transformation
is carried out according to Fig. 9.

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And, at last, fifth, let's execute 5-expansion which
eliminates distribution of computations BBSa and KKa
together with a clone of object a. The second copy of object
a cancels out its existence, and object S - begins the existence
in the environment. It is reflected in Fig. 10.

Now the desirable order of arranging the objects a, b and c
in the environment is reached, that is the objects c and b have
changed the places. The derived object, which is carrying out
such a rearrangement, we shall refer as combinator-permutator
and we shall denote it by C. From Fig. 10 it follows, that
C = S(BBS)(KK). As it has appeared, C exists, and its
characteristic is presented in Fig. 5.

14 Infinite constructions

Objects can lead to infinite constructions, and their interaction
has chained. Fig. 20 depicts a combinatory characteristic - the

Fig. 20. Synthesis for an object a the fixed point: Ya =

a(Ya) = a(a(Ya))

characteristic equality that needs to be added to a structure of
combinators to give a right to existence for combinator Y:

Ya = a(Ya),

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and this is a paradoxal combinator of H. B. Curry 2 . Combinator
Y makes more vivid rather uniform structure of objects and
combinators, giving rise to opportunity of organizing non-trivial
cyclic computations.

15 The plurality of the worlds of combinators

Attempts to answer the question, what - actually, - are the
constants often look boring enough, tiresome and burdened by
a set of more or less essential details. For the sake of justice
we shall note, that more often, from the resulted details their
most part appears not significant or, at least, not spilling greater
light. Possibly, in an area of computing, it will be possible to
receive missing details, and to look under other corner of sight
at the old ones. At least, the idea of constancy appears relative,
i.e. it is useful for considering not in a general universality, and
remaining within the limits of this or that system of computing.
We shall try to understand by an example, where it can appear
useful.

The set of objects together with their structuring
combinators looks rather homogeneous, even together with
opportunities for generation of cyclic constructions using the
combinator Y. It seems, that in similar structure there is no
place to any systems of objects with interesting and practically
significant behaviour. But this is not so.

Let's try to establish, whether will there be among objects
such an object V which is characterized by the distribution
relatively an application as follows:

V(ab)p = Vap(Vbp).

2 As known the object Y has a combinatory representation: Y = WS(BWB).
http://www.jurinfor.ru 43

Thus, all that is obtained is the structure of objects which is
enriched by this equality, and its action is illustrated in Fig. 21.

Fig. 21. Characteristic equality for object V: V(ab)p =
Vap(Vbp).

The kind of the left part of this equality suggests, that there
is a composition of the objects-maps V and a which is applied
to object-argument b, and the result of this application, in turn,
is applied to object p, playing a role of environment:

V(ab)p = ((V o a)(b))(p) = (V o a)bp = BVabp.

This mathematical idea is reflected in Fig. 22.

exp

@vab P ^=^ :' (v)©b P

red

Fig. 22. Canonical representation of the left side of the
characteristic equality of object V: V(ab)p = BVabp.

44

http://www.jurinfor.ru

Let's borrow now in the right part of the equality describing
behaviour of object V, and we shall try to receive its initial
representation. Fortunately, the most part of combinational
work on distribution of computations among objects incur
combinators W and <P, and corresponding reduction looks
laconically enough:

Vap(Vbp) = I(Vap)(Vbp) = <PI(Va)(Vb)p = &(<PI)Vabp

Corresponding transformations are depicted in Fig. 23.

Fig. 23. Canonical representation of the right side of the
characteristic equality of object V: Vap(Vbp) = &(<PI)Vabp.

For the purposes of the further ordering we shall merge the
executed transformations:

BVabp = V(ab)p = Vap(Vbp) = &(<PI)Vabp

The frame part of a chain of the derivation is presented in
Fig. 24.

What follows from the transformations above? It is possible
to receive several consequences.

First of all, it is easy to be convinced, that for V = K the
resulted construction works, so the object V does exist.

http ://www.jurinfor.ru

45

At the same time it is possible to write down mathematically
laconic sentence fixing the basically obtained result:

3V.Va,b,p.BVabp = V(<PI)Vabp.

®Vabp±=;( (V)0 N bp±=i:(y)ap (v)b p
red V y i red

Fig. 24. Resulting characteristic equality for object V: BVabp
= V(ab)p = Vap(Vbp) = V(<PI)Vabp.

This means that there is an object V such that for any objects
a, b, p the equality

B = V(&I),

is satisfied and characterizes the object V. Note that the object
p can be taken as an environment in the sense which is included
in this term in a theory of programming languages (see. [20];
[22], Ch. 12). The stated reasons lead to a construction resulted
in Fig. 25.

46

http://www.jurinfor.ru

Fig. 25. Combinatory characteristic generating the object V:

BV.Va, b, p.BVabp = V(<PI)Vabp.

Acknowledgements

Undoubtedly, the discussion of computing at the Conference
«Applicative computing systems» - ACS-2008, that was held in
April-May, 2008 at the Institute «JurInfoR-MGU», has served
as effective stimulus for some arranging the stated reasons.

Conclusions

Naturally, all of this is a fixing of some key ideas, but
development of plurality of the worlds of combinators
seems worthy and justifying efforts to preparation of a
preliminary textual material. The part from them has appeared
included in the Proceedings of ACS-2008, see URL http:
//jurinf or .exponenta. ru/ACS2008. Other part is
supposed to a consecutive statement.

References

1. Barendregt H., Wiedijk F. The Challenge of Computer Mathematics. -
Transactions of the Royal Society, Vol. 363, JVal835, 2005. - pp. 2351-2375.
ftp : //ftp .cs.ru. nl/pub/CompMath . Found/
Barendregt-Wiedi j k .pdf

http ://www.jurinfor.ru

47

2. Bell G., Dourish P. Yesterday's tomorrows: notes on ubiquitous computing's
dominant vision. - Personal Ubiquitous Comput., Vol. 11, N°2, 2007. - pp. 133—
143.

DOI http://dx.doi.org/10.1007/s0077 9-00 6-0 071-x.

3. Berners-Lee T., Hall W., Hendler J., O'Hara K., Shadbolt N., and Weitzner D. A
framework for Web science. - Foundations and Trends in Web Science, Vol. 1,
Issue 1,2006. -pp. 1-130.

http : //www. nowpublishers . com/web/

4. Cardelli L., Davies R. Service combinators for Web computing. - HP Labs
Technical Reports SRC-RR-148, June 1, 1997. - 15 p.

http: //www.hpl .hp. com/techreports /Compaq- DEC/
SRC-RR-148.html

5. Carpenter B. The Internet Engineering Task Force: Overview, Activities,
Priorities. - ISOC BoT, 2006-02-10, 2006.

http: //www. isoc . or g/ i so c /general/ trustees /docs/
Feb2006/IETF-BoT-20060210.pdf

6. Curry H.B. Functionality in combinatory logic. - Proc. National Academy of
Sciences of the USA, Vol. 20, 1934. - pp. 584-590.

7. de Bruijn N.G. Lambda-calculus notations with nameless dummies: a tool for
automatic formula manipulation. - Indag. Math. 1972, Ns34, pp. 381-392.

8. de Bruijn N. G. A survey of the project Automath. - In: To H.B. Curry: Essays
in combinatory logic, lambda calculus and formalism, Academic Press, 1980. -
pp. 579-606.

9. Denning P.J. Computing is a natural science. - Commun. ACM, Vol. 50, N°7,
2007. -pp. 13-18.

DOI http://dol.acm. org/10. 11 45/127251 6. 127252 9

10. Hindley J. R., Lercher B., Seldin J. P. Introduction to Combinatory Logic. -
London: Cambridge University Press, 1972.

11. Kennedy A. Functional Pearls: Pickler Combinators. - Journal of Functional
Programming, special issue on Functional Pearls, Vol. 14, N26, Cambridge
University Press, Nov 2004. - pp. 727-739,

12. MacLennan B.J. Molecular Combinator Reference Manual. - UPIM Report 2,
Technical Report UT-CS-02-489, Department of Computer Science, University
of Tennessee, Knoxville, 2002. - 16 p.

http: //www. cs .utk. edu/~mclennan/UPIM/CombRef .pdf

13. MacLennan B. J. Combinatory Logic for Autonomous Molecular Computation. -
Preprint of paper invited for Information Sciences, 2003.
http://www.cs.utk.edu/~mclennan/UPIM/CLAMC-IS.pdf

14. MacLennan B. J. Molecular Combinatory Computing for Nanostructure
Synthesis and Control. - IEEE Nano 2003, San Francisco, August 12-14, 2003.
http: //www. cs .utk. edu/~mclennan/UPIM/
MacLennan-MCCNSC .pdf

15. Schonfinkel M.I. Uber die Bausteine der mathematischen Logik. Math. Annalen
92, 1924. -pp. 305-316.

4o http://www.jurinfor.ru

16. Scott D. S. The lattice of flow diagrams. - Lecture Notes in Mathematics, 188,
Symposium on Semantics of Algorithmic Languages. - Berlin, Heidelberg, New
York: Springer-Verlag, 1971, pp. 311-372.

17. Scott D. S. Relating theories of the lambda calculus. - Hindley J., Seldin J. (eds.)
To H.B.Curry: Essays on combinatory logic, lambda calculus and formalism. -
N.Y.& L.: Academic Press, 1980, pp. 403-450.

18. Seldin J. P. The Logic of Curry and Church. - In: (Dov Gabbay and John
Woods, eds.) Handbook of the History of Logic, Vol. 5, Elsevier, 2006.
http : //people . uleth . ca/~ j onathan . seldin/CCL .pdf

19. Selinger P. and Valiron B. A lambda calculus for quantum computation with
classical control. - Mathematical Structures in Computer Science, 16(3), 2006. -
pp. 527-552.

20. BojitijieHrareH B.3. KoHcmpyKuuu n3biKoe npozpaMMupoeauua. HpueMbi onu-
canuR. - M.: AO «UeHrp K)pHH<])oP», 2001. - 276 c.

H3flaHne noflijepacaHO rpaHTOM POOH, npoeKT 01-01-14068-fl.

21. BojitrheHrareH B.3. KoAtSuHamopnaM nozuKa e npozpaMMupoeanuu. Bbmucjie-
nun c o6yeKmaMU e npuMepax u 3adauax. - M.: MHOH, 1994. - 204 c; 2-e
H3fl., M.: AO «HeHTp ±OpHH(J>oP», 2003. - 336 c.

22. BojitijieHrareH B.3. Memodbi u cpedcmea ebmucjienuu c o6yeKmaMU. AnnnuKa-
mueubie ebmucjiumejibubie cucmeMbi. - M.: JurlnfoR Ltd., AO «HeHTp K3pHH-
4>oP», 2004. - xvi+789 c.

H3flaHne noflijepacaHO rpaHTOM POOH, npoeKT 03-01-14055-fl.

23. HcMamioBa JLJO. JIozuKa o6-beKmoe. - B kh. [22], c. 613-630.

24. Kochkob OB. JIozuKa qbyHKUuoHajibnocmu. - B kh. [22], c. 595-612.

25. Kochkob OB. MnqbopMaifuoHnbie cucmeMbi: Kamezopubiu nodxod. - Hop, pefl.
JI.K). HcMaHJioBofi. - M.: «K)pHH(i>oP-npecc®», 2005. - 96 c.

26. IUayMaH OK. AnnjiuKamuenan zpajujwamuKa KaK ceManmuHecKaa meopua
ecmecmeenubix M3biKoe. - M.: HayKa, 1974. - 204 c.

http://www.jurinfor.ru 49

Index

a relativity

model

- values, 3

- computations, 2

combination

object

- identifier-value, 3

- constant, 6

combinator, 3

- contract, 8

computation

- derivative, 8

- environment, 3

- existing, 8

- model, 2

- generated, 8

— untyped, 5

- generation, 12

- result, 3

- initial, 8

computing, 1

- interaction, 7

- germination, 2

- redex, 8

- invariant, 2

- principle, 2

prestructure, 8

- revision of foundations, 5

proof

- sense of the term, 4

- as-object, 4

constancy

property

- property, 6

- of constancy, 6

constant

- definition, 3

range

- representation, 6

- definition, 3

- domain, 3

domain

- value, 3

- range, 3

rule, 4

- variable, 3

technology

environment

- information, 2

- computations, 3

theory

- interdisciplinary, 3

identifier

thinking

- value, 3

- computational, 2

knowledge

value

- possible, 4

- relativity, 3

- valid, 4

variable

- definition, 3

logic

- representation, 3

- combinatory, 3

- typed, 4

Part 2

Educational and methodical complex

of corresponding discipline

and ready-made solutions

Multimedia courses

The system of multimedia courses is intended to cover
the most complicated and important directions of modern
computing, computer science and information technologies.

♦ ♦ ♦

Combinatory logic

Course content. This course was delivered in NRNU MEPhI
and MIPT including 13 two hours lectures and reflects the
modern representations of fundamental basics of computing.
The main attention is paid to development the learners'
intuition which is sufficient to understand the granularity
of computations, its organizing using the block structure of
objects, the ability to construe the embedded computational
systems. In many cases the accent is given on development
the suitable embedded computational systems.

The course is equipped with the detailed textbooks with
a large amount of examples and tasks with the analysis
of solutions. An organization of books corresponds to the
multimedia course and allows to study the material in a variant
of «for the first reading» and does not demand the learner's

http://www.jurinfor.ru 5 1

preliminary background. In addition, as a result of studying
the course, a student acquires the knowledge of the primary
problems of computing which are at the frontier of modern
computer science.

♦ ♦ ♦

Wolfengagen V. E. Applicative Computa-
tional Technologies. Ready-made Solutions for
Engineer, Lecturer, Post-graduate and Graduate
Student. Edited by L. Yu. Ismailova. — Moscow:
«JurInfoR», 2009. — 64 p. (in Russian)

Summary. This work reflects the problems
of computing using the advanced and contemporary
mathematical means. Material is subdivided in two parts and
is approved in practice of teaching at NRNU «MEPhI», MIPT
and several other educational institutions of the country. The
1st part represents the outlook of computations, which is
achieved by the adoption of the atomistic doctrine for the
selected reference system of primary objects. This part is
intended for the advanced learners of discrete mathematics
(DM) and fundamental basics of information technologies
(FBIT), it is suitable for developing the intuition, which helps
to the successful navigation across the world with apparent
abundance and variety of the newly developed and ready-made
IT-solutions. The 2nd part gives the educational and methodical
complex of corresponding discipline (EMCD), equipped with
base textbooks, complemented by a series of monographs and
by media and environments for studying the computations with
the objects.

Material is intended for the instructor, the graduate student
and the student of IT-specialties and is the ready-made solution

52 http://www.jurinfor.ru

for the universities and the system of training and advancing
the qualification of specialists.

♦ ♦ ♦

.......... Wolfengagen V. E. The Functional Prog-

»yHKj;SSJ™{ioro ramming Paradigm. Edited by L.Yu. Ismailo-
va. — Moscow: «Center JurInfoR», 2010. — 80 p.

(in Russian)

Summary. This work covers the main
directions of evolving functional programming
ideas and technologies. This material is approved in practice of
NRNU MEPhI, MIPT and several other educational centers of
the Russian Federation.

Its 1st part represents an outlook of trends in usage
the pure functions in programming and is suitable both for
advanced learners and beginners in Computing and Information
Technologies.

The functional programming paradigm shown its great
importance and significantly influenced many branches of
computing and computer science. This is an evolving roadmap
having a perspective of growth in software engineering. It's
happened that program correctness is much easier to be
proved in case it is written in functional language. With
evolution of tools for supporting the proof this process seems
to become even easier. The functional programs can be treated
as executable specifications which themselves are already
prototypes, i.e. they do not need any additional efforts for
their preparing. The transformations of functional programs
are greatly simplified because of an algebraic origin of the
functions. Their usage opens the abilities for developing the
innovative machinery of the code optimization which grows up
the efficiency both of sequential and concurrent architectures.

http://www.jurinfor.ru 53

KOHCm'KUIIH

fBHKOB

JlPOrPAMMHPOBAHM

The amount of cases when an adoption of functional approach
for solving the real world tasks gives the obvious advantages
tends to growing up.

The 2nd part gives some supply of environments
for educational and methodical complex of corresponding
discipline (EMCD).

Material is intended for the instructor, postgraduate and
graduate students of IT-specialties.

♦ ♦ ♦

Wolfengagen V. E. The constructions of
programming languages. The techniques of
description. — Moscow: «Center JurInfoR» Ltd.,
2001. - 276 p. (in Russian)

Summary. This book covers the basics of
development, implementation and application of
constructions both of imperative and functional programming
languages. An essential attention is paid to application of
denotation semantics which allows in a complete degree
extracting the advantages of an object-oriented approach and
this, in turn, as a final score allows developing the target
computational model of purely functional kind.

The material is supplemented by the examples with detailed
explanations which are carefully commented assisting to study
the implementation of constructions from various languages.

This material can be used as a textbook or a guide. It
will be useful both for students and postgraduates, and for
professionals in computer science, information technologies
and programming.

RFBR's project 01-01-14068-fl.

♦ ♦ ♦

54 http://www.jurinfor.ru

KATJTOPHAJIlHAfl
AECTPAKTHAfl

MAIDHHA

Wolfengagen V. E. Categorical Abstract
Machine. Conspectus: Introduction to Com-
putations. — 2nd ed. — Moscow: «Center
JurInfoR», 2002. — 96 p. (in Russian)

Summary. This book contains the basics for
computation models in Computer Science. The
core topics both of lambda-calculus and combinatory calculi
are covered.

The main goals are to provide formal tools to assess
meaning of programming constructs in both a language-
independent and a machine independent way including
the code compiling and generation, its optimization and
runtime considerations. This is done using the categorical
abstract machine - relatively new direction in studying and
understanding the programs and computations. The material is
equipped with serial examples of growing up complexity.

This book is recommended to the undergraduate and
postgraduate students in Computer Science, Programming
Languages, Information Technologies, and Discrete Mathema-
tics.

♦ ♦ ♦

K0MEHHAT0PHA9
JIOFHKA

b nporPAMMnpoBAHUH

Wolfengagen V. E. Combinatory logic in
programming. Computations with objects
through examples and exercises. — 2-nd ed. —
Moscow: «Center JurInfoR», 2003. - VI+336 p.
(in Russian)

Summary. The book is intended for
computer science students, programmers and professionals
who have already got acquainted with the basic courses and

http://www.jurinfor.ru 55

background on discrete mathematics. It may be used as a
textbook for graduate course on theoretical computer science.

The book introduces a reader to the conceptual framework
for thinking about computations with the objects. The several
areas of theoretical computer science are covered, including
the following: type free and typed A-calculus and combinatory
logic with applications, evaluation of expressions, computations
in a category. The topics, covered in the book accumulated
much experience in teaching these subjects in graduate
computer science courses.

A rich set of examples and exercises, including solutions,
has been prepared to stimulate the self studying and to make
easier the job of instructor.

♦ ♦ ♦

,!„,.,. Wolfengagen V. E. Combinatory logic in

programming. Computations with objects
through examples and exercises. — 2-nd ed. —
Moscow: «Center JurInfoR», 2003. - X+336 p.
(in English)

Summary. The book is intended for
computer science students, programmers and professionals
who have already got acquainted with the basic courses and
background on discrete mathematics. It may be used as a
textbook for graduate course on theoretical computer science.

The book introduces a reader to the conceptual framework
for thinking about computations with the objects. The several
areas of theoretical computer science are covered, including
the following: type free and typed A-calculus and combinatory
logic with applications, evaluation of expressions, computations
in a category. The topics, covered in the book accumulated

56 http://www.jurinfor.ru

COMBCNATORY

IN PROGRAMMING

much experience in teaching these subjects in graduate
computer science courses.

A rich set of examples and exercises, including solutions,
has been prepared to stimulate the self studying and to make
easier the job of instructor.

♦ ♦ ♦

Wolfengagen V. E. Combinatory logic in
programming. Computations with objects
through examples and exercises. — 3-rd ed.,
revised. — Moscow: «Center JurInfoR», 2008. —
X+384 p. (in Russian)

KOMEHHATOPHAH

JlOrHKA

B nPOrPAMMHPOBAHHH

4?

Summary. The book is intended for
computer science students, programmers and professionals
who have already got acquainted with the basic courses and
background on discrete mathematics. It may be used as a
textbook for graduate course on theoretical computer science.

The book introduces a reader to the conceptual framework
for thinking about computations with the objects. The several
areas of theoretical computer science are covered, including
the following: type free and typed A-calculus and combinatory
logic with applications, evaluation of expressions, computations
in a category. The topics, covered in the book accumulated
much experience in teaching these subjects in graduate
computer science courses.

A rich set of examples and exercises, including solutions,
has been prepared to stimulate the self studying and to make
easier the job of instructor.

♦ ♦ ♦

http://www.jurinfor.ru 57

Wolfengagen V. E. Methods and Means
for Computations with Objects. Applicative
Computational Systems. — Moscow: JurlnfoR
Ltd., «Center JurlnfoR», 2004. - XVI+789 p.
(in Russian)

Summary. Those models, methods and
means are covered that are based on the notation of object. An
approach based on the operations of application and functional
abstraction is used resulting in the closed consideration of
applicative computations within the elementary framework. The
material covered in this book was used for delivering the
various versions of courses in computer science.

The essential theoretical background corresponds to the
high international level, and the basic computational ideas,
notions and definitions are explicated.

The book is intended for computer science students and
professionals in informatics. It may be used as a sourcebook
for graduate course on theoretical computer science.

RFBR's project 03-01-14055-/1.

♦ ♦ ♦

.*__ Wolfengagen V. E. Logic. Conspectus of

JIOrHKA ^ e l ectures i n formal reasoning. — 2nd
ed., completed and improved. — M.: «Center
JurInfoR» Ltd., 2004. - 229 p. (in Russian)

Summary. This edition is significantly
^^■>r;^^B improved and completed by the elements of
semantic reasoning using the classes and relations which are
especially important for working out the electronic forms of
information. The ways of reformulating the text of factual kind

5o http://www.jurinfor.ru

into symbolic language allowing the application of classic logic
means are considered. The techniques and ways of writing the
formal reasons and their validation procedure are shown. The
technique of formal logical reasonings, derivations and proofs
is illustrated by a lot of examples. The ways of including
the annotations (comments) into derivation are indicated which
allows checking the truth or false of reasons.

This book is primary intended for the students and
postgraduates of humanitarian specialties. It can be used for
the initial studying of the subject and for self studying as well.

♦ ♦ ♦

,.,„.._ Kosikov S.V. Information Systems: Catego-

m SS m ry Theory Approach. - Moscow: JurlnfoR-Press
Ltd., 2006. - 96 p. (in Russian)

Summary. The book gives an account to
the basic ideas related to the designing of
information systems by using methods of the
category theory. A detailed account is given of theoretico-
categorical constructions that are used for building theoretical
models and practical realization of information systems.

Considerable attention is given to how abstract machines
are built as a basis for realizing information systems by means
of category computational models.

The book can be used as reference material for students,
post-graduate students, experts in the field of designing
informational systems as well as for the application of
mathematical methods in the new information technology.

♦ ♦ ♦

http://www.jurinfor.ru 59

AnnjiHHan
BblHMCJlMTeflbHbie
CMCTeMbl

,n

The applicative computational systems.
Proceedings of the conference on applicative
computational systems (ACS'2008), Moscow:
April 29-30, 2008. /Dr. L.Yu. Ismailova (ed.). -
M.: NEI Institute of Contemporary Education
«JurInfoR-MGU», 2008. - 48 p. (in Russian)

Summary. Applicative computational systems, or ACSs
contain the calculi of objects based on Combinatory Logic
and lambda-calculus. The only thing which is essentially
developing in these systems is the representation of an object.
The combinatory logic contains the only metaoperator —
the application, or, in other terms, the action of one
object to another. The lambda-calculus contains a pair of
metaoperators — the application and functional abstraction
which allow binding of one variable within one object.

The objects which are generated in these systems are the
fundamental entities having the following properties: (1) the
number of argument places, or arity of an object is not fixed
from the beginning, but evolves step by step, in interaction with
other objects; (2) in developing the comprehended object one of
the initial objects — the function — is applied to other object —
its argument, but in other contexts they can change their roles,
i.e. both the functions and arguments are the objects on equal
rights; (3) a self application of the functions is allowed, i.e. an
object can be applied to itself. ACSs give the grounds for the
applicative approach to programming.

Applicative computing enables the combined development
of a computation as a relatively self contained block using the
existing blocks of computations, but all the variables in any
block of computation are bound and the block itself is closed.
The ACSs are used for enabling the applicative computing.

RFBR's project 08-07-06039-r.

60 http://www.jurinfor.ru

Contents

Part 1. Quarks, atoms, molecules of computing 3

Introduction 3

1 . Invariants of computing 6

2. New paradigms of computing 11

3. Revision of computing foundations 13

4. Notion of a constant 14

5. Functions 17

6. Interaction of an object and environment 17

7. The principles of computing 22

8. Interaction of objects 26

9. System of primary objects 27

10. System of derived objects 30

1 1 . Derivation of combinators 37

12. Reduction and expansion of objects 40

13. Synthesis of object with the given combinatory characteristic 41

14. Infinite constructions 42

15. The plurality of the worlds of combinators 43

Acknowledgements 47

Conclusions 47

References 47

Index 50

Part 2. Educational and methodical complex of corresponding discipline

and ready-made solutions 51

http://www.jurinfor.ru 61

Viacheslav Wolfengagen

Applicative computing

Its quarks, atoms and molecules

Technical editor A. E. Zaitsev

The production conforms the conditions of the

Public Health Department of the Russian Federation.

Sanitary- epidemiologic Certificate

N«- 77. 99.02. 953. D. 007462.11.05 issued by November 11, 2005

Signed to publishing 10.02.2010. Offset print. Offset paper.
Format 60x84/16. Pr. sh. 4. Copies 500. Order N«

"Center JurlnfoR" Ltd.
Phone (495) 778-87-26, 971-73-96,

http://www.jurinfor.ru, e-mail: info@jurinfor.ru

OTnenaTaHO b OAO «Illep6riHCKa$i Tnnorpa<pH5i»
117623, r. MocKBa, yji. TnnorpacbcKaa, fl. 10.

Notes

Notes